Calculating Net Electric Forces in a Charge Configuration

Here, we will walk through the process of calculating the net electric forces acting on charges located at specific points in space, using Coulomb's law. The charges are defined as follows:

Charge Configuration

A 100 μC charge is placed at the origin (0, 0). A -200 μC charge is located at (1 m, 1 m). A 400 μC charge is located at (0, 2 m).

Understanding Coulomb's Law

Coulomb's law is crucial in determining the electric force between two point charges. The formula for Coulomb's law is as follows:

F k frac{q_1 q_2}{r^2}

F is the electric force in newtons (N). k is Coulomb's constant, approximately 8.99 × 109 N m2/C2 q_1 and q_2 are the magnitudes of the charges in coulombs (C). r is the distance between the charges in meters (m).

Net Electric Force on the 100 μC Charge

Let's calculate the net electric force acting on the 100 μC charge at the origin.

Force due to the -200 μC Charge

Distance from the 0, 0 to 1, 1 is r_1 √(1^2 1^2) √2 ≈ 1.414 m Force due to the -200 μC charge is

F_1 8.99 × 109 × frac{100 × 10-6 × 200 × 10-6}{(1.414)^2} ≈ 89.9 N

Direction: Attractive towards the -200 μC charge, at an angle of 45° towards the 1st quadrant.

The components of the force are

F_{1x} -89.9 cos(45°) ≈ -63.5 N F_{1y} -89.9 sin(45°) ≈ -63.5 N

Force due to the 400 μC Charge

Distance from the 0, 0 to 0, 2 is 2 m. Force due to the 400 μC charge is F_2 8.99 × 109 × frac{100 × 10-6 × 400 × 10-6}{(2)^2} ≈ 89.9 N Direction: Repulsive away from the 400 μC charge in the positive y-direction.

The components of the force are

F_{2x} 0 N F_{2y} 89.9 N

Net Force on 100 μC Charge

Add the components to find the net force:

F_{netx} F_{1x} F_{2x} -63.5 N 0 N -63.5 N

F_{nety} F_{1y} F_{2y} -63.5 N 89.9 N ≈ 26.4 N

The net electric force on the 100 μC charge is:

F_{net} √((-63.5)^2 (26.4)^2) ≈ 69.6 N

Direction: Given by

θ tan-1(frac{26.4}{-63.5}) ≈ 202.5° from the positive x-axis

Net Electric Force on the -200 μC Charge

Now let's calculate the net electric force acting on the -200 μC charge.

Force due to the 100 μC Charge

Distance from the 0, 0 to 1, 1 is 1.414 m (same as above). Force due to the 100 μC charge is F_3 8.99 × 109 × frac{200 × 10-6 × 100 × 10-6}{(1.414)^2} ≈ 89.9 N Direction: Repulsive away from the 100 μC charge, at an angle of 45° towards the 2nd quadrant.

The components of the force are

F_{3x} 89.9 cos(45°) ≈ 63.5 N F_{3y} 89.9 sin(45°) ≈ 63.5 N

Force due to the 400 μC Charge

Distance from the 1, 1 to 0, 2 is 1 m. Force due to the 400 μC charge is F_4 8.99 × 109 × frac{200 × 10-6 × 400 × 10-6}{(1)^2} ≈ 719.6 N Direction: Attractive towards the 400 μC charge, in the negative y-direction.

The components of the force are

F_{4x} 0 N F_{4y} -719.6 N

Net Force on -200 μC Charge

Add the components to find the net force:

F_{netx} F_{3x} F_{4x} 63.5 N 0 N 63.5 N

F_{nety} F_{3y} F_{4y} 63.5 N - 719.6 N ≈ -656.1 N

The net electric force on the -200 μC charge is:

F_{net} √((63.5)^2 (-656.1)^2) ≈ 658.5 N

Direction: Given by

θ tan-1(frac{-656.1}{63.5}) ≈ -84.7° from the positive x-axis, or roughly downward

Summary of Results

Net Electric Force on 100 μC Charge: Magnitude: 69.6 N Direction: 202.5° from the positive x-axis Net Electric Force on -200 μC Charge: Magnitude: 658.5 N Direction: -84.7° from the positive x-axis, or roughly downward

Conclusion

By applying Coulomb's law and vector addition, we can understand and quantify the forces acting on point charges in a given space. This method is essential for advanced physics and engineering applications.